Optimal. Leaf size=162 \[ \frac {i 2^{-2 (m+3)} e^{4 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {4 i b (c+d x)}{d}\right )}{b}-\frac {i 2^{-2 (m+3)} e^{-4 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {4 i b (c+d x)}{d}\right )}{b}+\frac {(c+d x)^{m+1}}{8 d (m+1)} \]
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Rubi [A] time = 0.20, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4406, 3307, 2181} \[ \frac {i 2^{-2 (m+3)} e^{4 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,-\frac {4 i b (c+d x)}{d}\right )}{b}-\frac {i 2^{-2 (m+3)} e^{-4 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {4 i b (c+d x)}{d}\right )}{b}+\frac {(c+d x)^{m+1}}{8 d (m+1)} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3307
Rule 4406
Rubi steps
\begin {align*} \int (c+d x)^m \cos ^2(a+b x) \sin ^2(a+b x) \, dx &=\int \left (\frac {1}{8} (c+d x)^m-\frac {1}{8} (c+d x)^m \cos (4 a+4 b x)\right ) \, dx\\ &=\frac {(c+d x)^{1+m}}{8 d (1+m)}-\frac {1}{8} \int (c+d x)^m \cos (4 a+4 b x) \, dx\\ &=\frac {(c+d x)^{1+m}}{8 d (1+m)}-\frac {1}{16} \int e^{-i (4 a+4 b x)} (c+d x)^m \, dx-\frac {1}{16} \int e^{i (4 a+4 b x)} (c+d x)^m \, dx\\ &=\frac {(c+d x)^{1+m}}{8 d (1+m)}+\frac {i 4^{-3-m} e^{4 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {4 i b (c+d x)}{d}\right )}{b}-\frac {i 4^{-3-m} e^{-4 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {4 i b (c+d x)}{d}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 1.04, size = 213, normalized size = 1.31 \[ \frac {4^{-m-3} (c+d x)^m \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{-m} \left (-i d (m+1) \left (-\frac {i b (c+d x)}{d}\right )^m \left (\cos \left (4 a-\frac {4 b c}{d}\right )-i \sin \left (4 a-\frac {4 b c}{d}\right )\right ) \Gamma \left (m+1,\frac {4 i b (c+d x)}{d}\right )+i d (m+1) \left (\frac {i b (c+d x)}{d}\right )^m \left (\cos \left (4 a-\frac {4 b c}{d}\right )+i \sin \left (4 a-\frac {4 b c}{d}\right )\right ) \Gamma \left (m+1,-\frac {4 i b (c+d x)}{d}\right )+b 2^{2 m+3} (c+d x) \left (\frac {b^2 (c+d x)^2}{d^2}\right )^m\right )}{b d (m+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 134, normalized size = 0.83 \[ \frac {{\left (-i \, d m - i \, d\right )} e^{\left (-\frac {d m \log \left (\frac {4 i \, b}{d}\right ) - 4 i \, b c + 4 i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {4 i \, b d x + 4 i \, b c}{d}\right ) + {\left (i \, d m + i \, d\right )} e^{\left (-\frac {d m \log \left (-\frac {4 i \, b}{d}\right ) + 4 i \, b c - 4 i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {-4 i \, b d x - 4 i \, b c}{d}\right ) + 8 \, {\left (b d x + b c\right )} {\left (d x + c\right )}^{m}}{64 \, {\left (b d m + b d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{m} \cos \left (b x + a\right )^{2} \sin \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{m} \left (\cos ^{2}\left (b x +a \right )\right ) \left (\sin ^{2}\left (b x +a \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (d m + d\right )} \int {\left (d x + c\right )}^{m} \cos \left (4 \, b x + 4 \, a\right )\,{d x} - e^{\left (m \log \left (d x + c\right ) + \log \left (d x + c\right )\right )}}{8 \, {\left (d m + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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